Consider the following function:

$$f(x) = \frac{1}{\sqrt{x}}$$

As $x$ increases, the value of $f(x)$ decreases, but the decrease tapers off quickly as $x$ gets larger, and if you plot the graph of $f(x)$, the shape looks kind of like an upside-down logarithm. Would it be correct to describe this function as declining logarithmically, as $x$ increases?

  • $\begingroup$ No. The analogy is too superficial. $\endgroup$ – anon Dec 9 '12 at 6:19


Logarithmic functions are specifically functions that have the property that shifting the function up or down by a fixed amount is roughly equivalent to scaling it horizontally by some fixed factor. Your function $f(x) = x^{- \frac 12}$ is not the same in this regard.

For example, your function has a horizontal asymptote at x = 0. A logarithmic function has no horizontal asymptotes.

The function only "looks similar" because it is concave up for $x > 0$ and has a vertical asymptote at $x = 0$. The similarities stop there.


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